Solve for x |3x-5|<10

$| 3 x - 5 | < 10$

Step 1

Write $| 3 x - 5 | < 10$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$3 x - 5 \geq 0$

Solve the inequality.

Add $5$ to both sides of the inequality.

$3 x \geq 5$

Divide each term in $3 x \geq 5$ by $3$ and simplify.

Divide each term in $3 x \geq 5$ by $3$ .

$\frac{3 x}{3} \geq \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{5}{3}$

Divide $x$ by $1$ .

$x \geq \frac{5}{3}$

In the piece where $3 x - 5$ is non-negative, remove the absolute value.

$3 x - 5 < 10$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$3 x - 5 < 0$

Solve the inequality.

Add $5$ to both sides of the inequality.

$3 x < 5$

Divide each term in $3 x < 5$ by $3$ and simplify.

Divide each term in $3 x < 5$ by $3$ .

$\frac{3 x}{3} < \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{5}{3}$

Divide $x$ by $1$ .

$x < \frac{5}{3}$

In the piece where $3 x - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 x - 5) < 10$

Write as a piecewise.

${\begin{cases} 3 x - 5 < 10 & x \geq \frac{5}{3} \\ - (3 x - 5) < 10 & x < \frac{5}{3} \end{cases}$

Simplify $- (3 x - 5) < 10$ .

Apply the distributive property.

${\begin{cases} 3 x - 5 < 10 & x \geq \frac{5}{3} \\ - (3 x) - - 5 < 10 & x < \frac{5}{3} \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 3 x - 5 < 10 & x \geq \frac{5}{3} \\ - 3 x - - 5 < 10 & x < \frac{5}{3} \end{cases}$

Multiply $- 1$ by $- 5$ .

${\begin{cases} 3 x - 5 < 10 & x \geq \frac{5}{3} \\ - 3 x + 5 < 10 & x < \frac{5}{3} \end{cases}$

Step 2

Solve $3 x - 5 < 10$ when $x \geq \frac{5}{3}$ .

Solve $3 x - 5 < 10$ for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Add $5$ to both sides of the inequality.

$3 x < 10 + 5$

Add $10$ and $5$ .

$3 x < 15$

Divide each term in $3 x < 15$ by $3$ and simplify.

Divide each term in $3 x < 15$ by $3$ .

$\frac{3 x}{3} < \frac{15}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{15}{3}$

Divide $x$ by $1$ .

$x < \frac{15}{3}$

Simplify the right side.

Divide $15$ by $3$ .

$x < 5$

Find the intersection of $x < 5$ and $x \geq \frac{5}{3}$ .

$\frac{5}{3} \leq x < 5$

Step 3

Solve $- 3 x + 5 < 10$ when $x < \frac{5}{3}$ .

Solve $- 3 x + 5 < 10$ for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Subtract $5$ from both sides of the inequality.

$- 3 x < 10 - 5$

Subtract $5$ from $10$ .

$- 3 x < 5$

Divide each term in $- 3 x < 5$ by $- 3$ and simplify.

Divide each term in $- 3 x < 5$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} > \frac{5}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} > \frac{5}{- 3}$

Divide $x$ by $1$ .

$x > \frac{5}{- 3}$

Simplify the right side.

Move the negative in front of the fraction.

$x > - \frac{5}{3}$

Find the intersection of $x > - \frac{5}{3}$ and $x < \frac{5}{3}$ .

$- \frac{5}{3} < x < \frac{5}{3}$

Step 4

Find the union of the solutions.

$- \frac{5}{3} < x < 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{5}{3} < x < 5$

Interval Notation:

$(- \frac{5}{3}, 5)$

Step 6

Solve for x |3x-5|<10

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